1. Quadratic Forms and Objective functions with two or more variables

2. Two Choice Variables Quadratic Forms Definiteness
Let 𝑢=𝑑𝑥, 𝑣=𝑑𝑦, 𝑎= 𝑓 𝑥𝑥 , 𝑏= 𝑓 𝑦𝑦 , ℎ= 𝑓 𝑥𝑦 = 𝑓 𝑦𝑥
Then 𝑑 2 𝑍= 𝑓 𝑥𝑥 𝑑 𝑥 2 +2 𝑓 𝑥𝑦 𝑑𝑦𝑑𝑥+ 𝑓 𝑦𝑦 𝑑 𝑦 (𝑒𝑞𝑛 1)
can be written as 𝐝 𝟐 𝐙=𝒒=𝒂 𝒖 𝟐 +𝟐𝒉𝒖𝒗+𝒃 𝒗 𝟐 (𝑒𝑞𝑛 2) 
Definiteness
Positive definite if q is invariably positive (q > 0)
Positive semidefinite if q is invariably nonnegative (𝑞≥0 )
Negative definite if q is invariably negative (q < 0)
Negative semidefinite if q is invariably non positive (𝑞≤0 )

3. Rewriting the quadratic form using completing square
𝒂 𝒖+ 𝒉 𝒂 𝒗 𝟐 +𝒂𝒃− 𝒉 𝟐 𝒂 𝒗 𝟐
𝑞 𝑖𝑠 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑑𝑒𝑓𝑖𝑛𝑖𝑡𝑒 𝑖𝑓𝑓 𝑎>0 𝑎𝑛𝑑 𝑎𝑏− ℎ 2 >0
𝑞 𝑖𝑠 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑑𝑒𝑓𝑖𝑛𝑖𝑡𝑒 𝑖𝑓 𝑎<0 𝑎𝑛𝑑 𝑎𝑏− ℎ 2 >0
Writing in Matrix Form
We have 𝑢 𝑣 𝑎 ℎ ℎ 𝑏 𝑢 𝑣
The determinant of the coefficient matrix 𝑎 ℎ ℎ 𝑏 is important in determining the sign of 𝑞
𝑞 𝑖𝑠 𝒑𝒐𝒔𝒊𝒕𝒊𝒗𝒆 𝒅𝒆𝒇𝒊𝒏𝒊𝒕𝒆 𝑖𝑓𝑓 ∣𝑎∣>0 𝑎𝑛𝑑 𝑎 ℎ ℎ 𝑏 >0
𝑞 𝑖𝑠 𝒏𝒆𝒈𝒂𝒕𝒊𝒗𝒆 𝒅𝒆𝒇𝒊𝒏𝒊𝒕𝒆 𝑖𝑓𝑓 ∣𝑎∣<0 𝑎𝑛𝑑 𝑎 ℎ ℎ 𝑏 >0

4. Writing equation (2) in terms of equation (1) gives further insights
Since 𝑎= 𝑓 𝑥𝑥 𝑎𝑛𝑑 𝑎𝑏− ℎ 2 𝑖𝑠 𝑓 𝑥𝑥 𝑓 𝑦𝑦 − 𝑓 𝑥𝑦 . 𝑓 𝑦𝑥 𝑓 𝑥𝑥 𝑓 𝑦𝑦 − 𝑓 𝑥𝑦 . 𝑓 𝑦𝑥 𝑜𝑟 ( 𝑓 𝑥𝑥 𝑓 𝑦𝑦 − 𝑓 𝑥𝑦 2 )
Conditions for 𝑑 2 𝑍 𝑓𝑜𝑟 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑑𝑒𝑓𝑖𝑛𝑖𝑡𝑒 𝑖𝑠 𝑓 𝑥𝑥 > 0 𝑎𝑛𝑑 𝑓 𝑥𝑥 𝑓 𝑥𝑦 𝑓 𝑥𝑦 𝑓 𝑦𝑦 >0
𝑑 2 𝑍 𝑓𝑜𝑟 𝑛𝑒𝑔𝑖𝑡𝑖𝑣𝑒 𝑑𝑒𝑓𝑖𝑛𝑖𝑡𝑒 𝑖𝑠 𝑓 𝑥𝑥 <0 𝑎𝑛𝑑 𝑓 𝑥𝑥 𝑓 𝑥𝑦 𝑓 𝑥𝑦 𝑓 𝑦𝑦 >0

5. Discriminant of a Quadratic form
In general the discriminant of a quadratic form
𝑞=𝑎 𝑢 2 +2ℎ𝑢𝑣+𝑏 𝑣 2
Is the symmetric determinant 𝑎 ℎ ℎ 𝑏
In the particular case of quadratic form
𝑑 2 𝑍= 𝑓 𝑥𝑥 𝑑 𝑥 2 +2 𝑓 𝑥𝑦 𝑑𝑦𝑑𝑥+ 𝑓 𝑦𝑦 𝑑 𝑦 2
The discriminant is the determinant with second order partial derivatives as its elements.

6. Hessian Determinant
∣𝐻∣= 𝒇 𝒙𝒙 𝒇 𝒙𝒚 𝒇 𝒚𝒙 𝒇 𝒚𝒚
Determinant with all the second order partial derivatives is called Hessian Matrix.

7. Extremum Conditions for two choice variable
𝑖𝑓 𝑑 2 𝑍 𝑖𝑠 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑑𝑒𝑓𝑖𝑛𝑖𝑡𝑒 𝑖𝑡 𝑖𝑠 𝑚𝑖𝑛𝑖𝑚𝑢𝑚
𝑖𝑓 𝑑 2 𝑍 𝑖𝑠 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑑𝑒𝑓𝑖𝑛𝑖𝑡𝑒 𝑡ℎ𝑒𝑛 𝑚𝑎𝑥𝑖𝑚𝑢𝑚
So,
𝑑 2 𝑍 𝑓𝑜𝑟 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑑𝑒𝑓𝑖𝑛𝑖𝑡𝑒(𝑚𝑖𝑛𝑖𝑚𝑢𝑚) 𝑖𝑓 𝑓 𝑥𝑥 > 0 𝑎𝑛𝑑 𝑓 𝑥𝑥 𝑓 𝑥𝑦 𝑓 𝑥𝑦 𝑓 𝑦𝑦 >0
𝑑 2 𝑍 𝑓𝑜𝑟 𝑛𝑒𝑔𝑖𝑡𝑖𝑣𝑒 𝑑𝑒𝑓𝑖𝑛𝑖𝑡𝑒 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑖𝑓 𝑓 𝑥𝑥 < 0 𝑎𝑛𝑑 𝑓 𝑥𝑥 𝑓 𝑥𝑦 𝑓 𝑥𝑦 𝑓 𝑦𝑦 >0

8. Objective Functions with 3 choice Variables
𝑧=𝑓 𝑥 1 , 𝑥 2 , 𝑥 3
𝑑𝑧= 𝑓 1 𝑑 𝑥 1 + 𝑓 2 𝑑 𝑥 2 + 𝑓 3 𝑑 𝑥 3
First order condition: 𝑑𝑧=0, 𝑖.𝑒. 𝑓 1 = 𝑓 2 = 𝑓 3 =0
Second order condition : 𝑑 2 𝑍
𝑑 2 𝑍= 𝜕 𝜕 𝑥 1 𝑓 1 𝑑 𝑥 1 + 𝑓 2 𝑑 𝑥 2 + 𝑓 3 𝑑 𝑥 3 𝑑 𝑥 1 + 𝜕 𝜕 𝑥 2 𝑓 1 𝑑 𝑥 1 + 𝑓 2 𝑑 𝑥 2 + 𝑓 3 𝑑 𝑥 𝑓 1 𝑑 𝑥 1 + 𝑓 2 𝑑 𝑥 2 + 𝑓 3 𝑑 𝑥 3 𝑑 𝑥 2 + 𝜕 𝜕 𝑥 3 𝑓 1 𝑑 𝑥 1 + 𝑓 2 𝑑 𝑥 2 + 𝑓 3 𝑑 𝑥 3 𝑑 𝑥 3
= 𝑓 11 𝑑 𝑥 𝑓 12 𝑑 𝑥 1 𝑑 𝑥 2 + 𝑓 13 𝑑 𝑥 1 𝑑 𝑥 3 + 𝑓 21 𝑑 𝑥 2 𝑑 𝑥 1 + 𝑓 22 𝑑 𝑥 𝑓 23 𝑑 𝑥 2 𝑥 3 + 𝑓 31 𝑑 𝑥 3 𝑑 𝑥 1 + 𝑓 32 𝑑 𝑥 3 𝑑 𝑥 2 + 𝑓 33 𝑑 𝑥 3 2

9. Hessian Matrix for 3 choice Variables
𝐻= 𝑓 11 𝑓 12 𝑓 13 𝑓 21 𝑓 22 𝑓 23 𝑓 31 𝑓 32 𝑓 33
𝑤ℎ𝑒𝑟𝑒 ∣ 𝐻 1 ∣= 𝑓 11 , ∣ 𝐻 2 ∣= 𝑓 11 𝑓 12 𝑓 21 𝑓 𝑎𝑛𝑑 ∣ 𝐻 3 ∣= 𝑓 11 𝑓 12 𝑓 13 𝑓 21 𝑓 22 𝑓 23 𝑓 31 𝑓 32 𝑓 33
Optimization Conditions
First Order Necessary condition: 𝒇 𝟏 = 𝒇 𝟐 = 𝒇 𝟑 =𝟎
Second order Conditions
𝑓𝑜𝑟 𝑍 ∗ 𝑡𝑜 𝑏𝑒 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 ∣ 𝐻 1 ∣<0, ∣ 𝐻 2 ∣>0, ∣ 𝐻 3 ∣<0 ; 𝑑 2 𝑍(𝑁𝐷)
𝑓𝑜𝑟 𝑍 ∗ 𝑡𝑜 𝑏𝑒 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 ∣ 𝐻 1 ∣>0, ∣ 𝐻 2 ∣>0, ∣ 𝐻 3 ∣>0 ; 𝑑 2 𝑍(𝑃𝐷)

10. 𝑵 Variable Case First order Necessary conditions
Optimization Conditions
First order Necessary conditions
𝑓 1 = 𝑓 2 = 𝑓 3 = 𝑓 4 =…………… 𝑓 𝑛 =0
Second order Conditions
𝑓𝑜𝑟 𝑍 ∗ 𝑡𝑜 𝑏𝑒 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 ∣ 𝐻 1 ∣<0, ∣ 𝐻 2 ∣>0, ∣ 𝐻 3 ∣< 0,∣ 𝐻 4 ∣>0… −1 𝑛 ∣ 𝐻 𝑛 ∣> ; 𝑑 2 𝑍(𝑁𝐷)
𝑓𝑜𝑟 𝑍 ∗ 𝑡𝑜 𝑏𝑒 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 ∣ 𝐻 1 ∣>0, ∣ 𝐻 2 ∣>0, ∣ 𝐻 3 ∣> 0 ….∣ 𝐻 𝑛 ∣>0; 𝑑 2 𝑍(𝑃𝐷)